Assigning a value to a power likelihood in a general Bayesian model

Bayesian approaches to data analysis and machine learning are widespread and popular as they provide intuitive yet rigorous axioms for learning from data; see Bernardo and Smith (2004) and Bishop (2006). However, this rigour comes with a caveat that the Bayesian model is a precise reflection of Nature. There has been a recent trend to address potential model misspecification by raising the likelihood function to a power, primarily for robustness reasons, though not exclusively. In this paper we provide a coherent specification of the power parameter once the Bayesian model has been specified in the absence of a perfect model.